## Abstract

In this paper, we report the formation of extremely sharp (Quality factor *Q*~ + ∞) FR in a single layer of dielectric nanorods with perturbed periodicity. The interference between the broadband Fabry-Perot (F-P) resonance and defect induced dark mode results in refractive index sensitivity (*S*) of 1312.75 nm/RIU and figure of merit (*FOM*) of 500, offering an excellent platform for biological sensing and detection.

© 2015 Optical Society of America

## 1. Introduction

Metamaterial is a new class of artificial materials possessing exotic electromagnetic characteristics that do not exist in nature. For example, perfect absorption [1], selective emission [2], and the electromagnetically induced transparency-like effect [3] have attracted much research attention. To enhance light-metamaterial interactions at a very fundamental level, a great deal of applications requires sharp resonance accompanying high *Q*-factor combined with large field enhancement. However, the ohmic loss of metals and surface roughness induced by imperfect fabrication limit the *Q*-factor in metamaterials especially at optical frequencies. Meanwhile, the field strongly coupled to surroundings causes high radiative loss which usually existed in metal-dielectric periodic structures. To relax such crucial tasks for realistic applications, dielectric metamaterials that could support Fano resonances (FR) offer potential solutions.

As an analogy of quantum interference between continuum with discrete state in atomic physics, asymmetric line shape Fano resonance has been intensively studied [4–11]. In all-dielectric system, the material loss is not an insurmountable problem anymore. Actually, one should note that photonic crystals have been extensively studied by various groups and Fano resonance is attributed to either guided resonance or diffraction mode (Rayleigh anomalies). For example, Fan et al gave an temporal coupling model of Fano resonance in photonics crystals slab and demonstrated that the *Q*-factor of the resonance is strongly influenced by the symmetry of the modes and the radius of the holes [12,13]. In 2013, they proposed higher *Q* filters based on F-P resonances obtained in double-layer photonic crystals structures with an optimized *Q* value of 22000 by design and experimentally demonstrated *Q* close to 10000 [14].

In the previous works, we reported Fano resonance induced by analogous quadrupole coupling in a cylinder bars array. High *Q*-factor (~18000) and large field enhancement (> 10000) have been theoretically predicted [15]. Unfortunately, ultrahigh Q-factor seems difficulty to obtain in such periodic grating structures since the existence of the leaky mode.

It is shown recently that ultrahigh Q-factor induced by sharp trapped mode can be obtained in planar metamaterial by breaking the symmetry [16,17]. Different lengths of two parallel ceramic wires lead to interference between the quadrupole and dipole modes. *Q*-factor as large as 62 is observed which is comparable to the asymmetrically split rings structure [17] at microwave region.

In this paper, we study Fano resonance in a single layer of silicon nanorods. As the distance of adjacent resonators reaches a critical value, an extremely sharp (*Q*~36000) resonance takes place due to the analogous quadrupole mode. It is then numerically demonstrated that ultrahigh *Q*-factor (~5*10^{8}) can be obtained with perturbed periodicity, which offers a trapped mode to create artificial electromagnetic response and couple weakly with the external field. Furthermore, it is predicted that infinitely high *Q*-factor can be theoretically achieved in the single layer structure. The gap region is adaptive for applications such as microfluidic channel in label-free biological sensing.

## 2. Structure and simulation

Figure 1 shows the conceptual design of the simulating process. Transverse magnetic (TM) planewave feeds the single layer of silicon nanorods and the deep color shows the reflecting /transmitting light with FR. As depicted in the amplifying zone, the rectangular silicon resonator array is formed on the top of quartz substrate. The width and height of the nanorods are indicated as *w _{n}* (

*n*= 1, 2, 3) and

*h*, respectively. The period of the structure keeps

*p*= 600 nm. The refractive indices of silicon and quartz are 3.46 and 1.48 [18] in the frequency range investigated here (The material loss will be considered later). A finite element method (FEM) based solver CST 2013 is adopted to analyze the unit cell while periodic boundary conditions are used along

*x*and

*y*directions.

#### 2.1 FR in the symmetric structure

For clarity of the discussion, the widths of the resonators are set equal (*w _{1}* =

*w*=

_{2}*w*) and the substrate is not considered here. The transmission and reflection (

_{3}= w*t*=

*S*and

_{21}*r*=

*S*) for different nanorods widths are illustrated in Fig. 2(

_{11}*w*= 370 nm, 400 nm, 420 nm and 450 nm). The other geometric parameters are chosen as

*p*= 600 nm and

*h*= 500 nm. Obviously, the linewidth of the resonance reaches the minimum at

*w*= 420 nm, where the corresponding

*Q*-factor can be calculated as

*Q*=

*f/Δf*= 215.9775/0.006 = 35996 (

*f*is a resonant frequency of the FR and

*Δf*is defined as the difference of the frequency at transmission peak and dip). Interestingly, either a smaller or larger resonater width will decrease the

*Q*-factor.

In fact, F-P resonance can occur for a dielectric slab when the refractive index is high enough. The rod array can be treated as an equivalent slab according to the efficient medium theory when the period is much smaller than the wavelength. In order to make it clear, the transmittance and reflectance of a dielectric slab and a rod array are shown as Fig. 3(a) and 3(b) for comparison. Figure 3(a) indicates that the F-P resonance is caused by the interference of multiple reflections in the two interfaces of the slab. The zoom-in plots of transmission and reflection spectra of the single layer of nanorods possess abrupt change between peaks and dips when *w* is 420 nm as depicted in Fig. 3(c). Using the temporal coupled mode theory [13], the intensity reflection coefficient *R* can be written as:

Where *r* and *t* are the reflection and transmission spectra due to broadband F-P resonance, *ω _{0}* indicates the resonance frequency and

*γ*represents the damping coefficient caused by radiative loss. In order to get more physical insight into this extremely sharp resonance, we calculated and fitted the transmission spectra according to the formula (1), depicted as the blue curve in Fig. 3(c). When suitably evaluated the

*ω*and

_{0}*γ*, the analytical mode fits well with the FEM simulation results. From the fitting parameters,

*ω*= 2π*215.9755 THz and

_{0}*Q*=

*ω*37694 have been obtained, which agrees well with the values calculated according to the Fig. 2.

_{0}/γ =Figure 3(d) shows the electric field distribution along *x* direction corresponding to the maximum *Q*-factor at resonance frequency 215.9775 THz. Evidently, the waveguide mode is an ensemble of analogous quadrupole coupling. Since the outgoing waves on the top of the resonators possess the same phase with that distributed below the rod, the sign of ± should be chosen as plus and the FR in this structure is induced by interference between the broadband F-P resonance and the analogous quadrupole modes. The antiphase oscillations caused by the reversed currents result in the reduction of loss in the FR. Therefore the quadrupole mode has a much smaller radiative decay than the dipole mode, which had been experimentally demonstrated in the asymmetric double bars [19]. Of course, the current discussed here becomes displacement current, rather than conducting current. As depicted in Fig. 3(d), the maximal electric field (*|E| _{max}*) reaches about 200 V/m and the corresponding intensity is 40000 times larger than the incident field (set as 1 V/m).

#### 2.2 FR in the perturbed periodic structure

In recent decades, some groups have expended a great deal of efforts to investigate the narrow bandwidth filter [20] and sensitive bio-detection [21]. The central concern is mostly minimizing the radiative loss to maximize the *Q*-factor. This is a challenge when the metal induced ohmic loss or in a dielectric system with an incomplete band gap [22]. However, some high performance structures have been designed and cited in this paper, such as a sharp trapped mode with ultrahigh *Q*-factor in metamaterial by introducing symmetry breaking in the shape of elementary structure [23,24]. To consider this factor, different widths of the resonators are redesigned to perturb the periodicity. In this case, three adjacent resonators actually form a new unit cell. In order to keep the feature sizes at subwavelength range, the frequencies investigated are scaled down to ~91 THz for the modified period 180 nm. Little change [18] in the refractive indices of the materials can be neglected. The thickness of the resonators is set as *h* = 200 nm and the substrate is thick enough to support the silicon nanorods. We fixed the *w _{1}* and

*w*as 440 nm, varying

_{3}*w*

_{2}to plot the transmission spectra as shown in Figs. 4(a)-4(f). ∆

*w*=

*w*

_{2}-

*w*

_{1}has been used to describe the degree of the perturbation.

*Q*-factor increases monotonically as the ∆

*w*reduces. However, the gray curve in Fig. 4(a) indicates that the ultrasharp resonance disappears when the ∆

*w*is 0 nm.

In this single layer of nanorods design, quality factor *Q* of 200000 and 550000 can be theoretically achieved with *w*_{2} of 445 nm and 443 nm, respectively. A much higher *Q*-factor of 4.5*10^{6} can be obtained when ∆*w* is 1 nm. This ultrafine structure at infrared region is compatible with the present nanofabrication technology. For example, a slit as small as 1 nm had been realized by atomic-layer deposition last year [25]. With the reduction of the degree of the perturbation, all resonances including the sharp FR resonance shift towards higher frequencies and nearly 2*10^{7} has been theoretically achieved for ∆*w* = 0.5 nm as depicted in Fig. 5(a). Surprisingly, the *Q*-factor can be still enhanced to 5*10^{8} when the discrepancy between the adjacent resonators is 0.1 nm. It means that infinite *Q*-factor will be excited in this perturbed periodic nanorods as long as the width of the middle resonator (*w*_{2}) tends to that of the adjacent one (*w*_{1}) in the perturbed elementary structure. Notably, this is different from previously reported sharp resonance in perforated dielectric slabs [13,14], where the maximum *Q*-factor corresponds to a vanishing hole diameter.

Simulated field distribution profiles for ∆*w* = 0 nm, 0.1 nm and 10 nm at corresponding resonance frequency have been plotted in Fig. 5(b). Our calculations showed that in the case of symmetric structure, the incident light transmits through the structure, causing the disappearance of the Fano line shape. When ∆*w* is 0.1 nm, as depicted in the middle plot of the Fig. 5(b), two parts of the perturbed unit cell are excited in antiphase, while fields have almost the same amplitude. It is shown that the electromagnetic wave is confined in the structure and forms a standing wave. The scattered electromagnetic fields produced by such modes are very weak, which dramatically reduces coupling to free space and therefore radiation losses.

The origin of the unusually sharp spectral responses of the nanorods array can be reckoned as the trapped modes which are weakly coupled to free space. It is this property of the trapped modes and the lossless dielectric material that allows in underlying mechanisms to achieve ultrahigh *Q*-factor resonances in such thin structures (*h* = 200 nm). These modes are inaccessible in the symmetric grating designed in this section, but can be excited if the nanorods array has certain structural asymmetry that allows weak coupling to free space. The perturbation in the widths of the resonators in this structure disturbs the discrete translational symmetry and forms a narrowband dark mode to interfere with the broadband F-P resonance. Furthermore, the amplitudes of the fields will increase on reducing the degree of the asymmetry. The trapped mode excited when ∆*w* is 0.1 nm is significantly larger than the resonance occurred when ∆*w* is 10 nm, which yields higher *Q*-factors for this type of the response.

However, in a practical system, the achievable field enhancement is limited by the finite number of unit cells. This can be calculated by COMSOL Multiphysics 4.3 with perfect matched layers (PML) boundary conditions. The maximal electric fields for rods array with 20 unit cells and 40 unit cells along *y* axis (length in *x* axis is set as infinite long) are calculated with intensity enhancement of 300 and 9000. This spectral collapse effect can be attributed to the coherent characteristic of such Fano resonance [26].

#### 2.3 Application in the biological sensors

In this section, the structure is investigated as a biological sensor by filling the gap region with liquid solution. The loss tangent of liquid is neglected and the *w*_{2} is set as 450 nm considering the possibility of fabrication. As shown in Fig. 6, the line shape of transmission becomes more asymmetric due to the influence of the liquid. Obviously, a rather small change in the refractive index of the solution can result in a resolvable spectral tuning of transmission. The extremely large sensitivity stems from the fact that the *Q*-factor is very high and a majority of energy is concentrated in the liquid solution. In order to evaluate the performance of bio-sensing, the sensitivity (*S*) and figure of merit (*FOM*) are calculated using [27]:

From Fig. 6, one can calculate that *S* = 1312.75 nm/RIU and *FOM* = 500, which is much larger than the previous reported results obtained in plasmonic structure [27], dielectric system [28] and metal nanorod dimer [29].

#### 2.4 Influence of loss

In the above discussion, the material loss is neglected since the silicon and the quartz in the frequency region investigated can be considered as lossless dielectric. However, the performance of the FR is greatly affected by the loss of the material and such influence could not be avoided in the application. According to [18], the loss tangent of 8.7*10^{−11} is added in the silicon. As depicted in Fig. 7, extremely sharp resonance is still observed when *w _{2}* is 440.1 nm. Nevertheless, the

*Q*-factor and the difference between transmission values at the peak and dip of the resonance decrease slightly. Then, the permittivity is set with a larger loss tangent of 8.7*10

^{−8}as the blue curve describes which shows that the phenomenon of FR is fainter when the loss increases.

## 3. Conclusions

In summary, we demonstrated theoretically and numerically that sharp Fano resonance can be excited in a single layer of nanorods with perturbed periodicity. Based on the interference between the broadband F-P resonance and the strongly coupled analogous quadrupole mode or trapped mode, ultrahigh *Q*-factor (infinite in theory) and large field enhancement have been obtained. Since the maximal *Q*-factor corresponds to a rather large gap width, the gap region between adjacent nanorods forms a wide space for applications such as biological sensors. The structure can also be scaled to other frequency bands such as terahertz and microwave frequencies due to the scalability of Maxwell’s equations.

## Acknowledgments

This work was supported by 973 Program of China (No. 2013CBA01700), National Natural Science Funds (No. 61138002).

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